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1. (Band-pass filter) Comsider the operational amplifier circuit in Figure 1. Assuming the operational amplifier is ideal. find a differential equation relating the input voltage signal x with the output voltage signal g- Find the transfer function of the system H mapping x to g- Find the poles and zeros of the system and construct a pole-zero plot. Assert that H is stable and regular and find and plot its impulse response h. Build the circuit om a breadboard and mensure the spectrum of the circuit by using the soundcard to input sinusoidal signals of the form = 110 le = 0,1, 12. For oach k = 0,1 12 obtain an estimate of the spectrum aH(fk). You may wish to use the method described in Test 4 of the lecture notes. Find an analytical expression for the spectrum of the system AH(f) and plot the magnitude and phase spectrum over the interval f € [0, 7300j. At what positive frequency f is the magaitude spectium maximised? On the same plota draw the mensurements of the magaitude and phase spectrum obtained using the sound card Assert that the mensurements conform with the hypothesised spectrum aH(f). List the components used in comstructing the circuit. Now input the signal =(t)=}sin(2sf1t)+}sin(2xf2t) with fi = 500 and f2 = 1333. Using the soundcard simultaneously record the input signal x and also the output voltage signal y- Build reconstructed approximate signals il and @ from the samples obtained. Plot: i. @ and Hi= over a 4ms duration. Assert that h*i is close to g. To compute h* you may wish to use the trapenoidal integration method used in Test 5 of the lecture notes. 2. (Butterworth filter) Design a lowpass second order Butterworth filter with cuttoff frequency in the range 2200Hz to 2600Hz. Draw a diagram of the electrical circuit you have designed and list the compo- nents. Derive the transfer function and the spectrum of your filter. Construct the circuit and, using the computer soundcard, measure its spectrum over frequencies in the range 100 Hz to 7000 Hz. Plot your measurements alomgide the hypothesised spectrum that you derived. 10nF 15k? 100 nF 33000 4i(t) =(1) @(i) Figure 1: Operational amplifier configured as a band-pass filter with two capacitors and two resistors.

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